This statement can be derived from first principles by noting that under certain conditions, the function Z depends only on the thermodynamic beta to the minus-one-half, and evaluating the derivative of the log of that function with respect to beta.  Debye's theory of heat capacity reduces to the Du
This statement can be derived from first principles by noting that under certain conditions, the function Z depends only on the thermodynamic beta to the minus-one-half, and evaluating the derivative of the log of that function with respect to beta.  Debye's theory of heat capacity reduces to the Dulong-Petit law in the high-temperature limit as a direct consequence of this statement. The truth of this statement depends on the truth of the (*) ergodic hypothesis, and this statement is a direct consequence of the fact that in a Maxwell-Boltzmann distribution, the numerator of the exponent is the kinetic energy of the molecule. For ten points, identify this theorem which at equilibrium allocates energy equal to one-half times Boltzmann's constant times the temperature for every quadratic degree of freedom.
This statement can be derived from first principles by noting that under certain conditions, the function Z depends only on the thermodynamic beta to the minus-one-half, and evaluating the derivative of the log of that function with respect to beta.  
{equipartition} theorem